EST II MATHEMATICS — LEVEL 1 HARD PRACTICE MOCK EXAM — 2026 SPECIFICATIONS 60 Questions | No Geometry | No Pictures

 

EST II MATHEMATICS — LEVEL 1

HARD PRACTICE MOCK EXAM — 2026 SPECIFICATIONS
60 Questions   |   No Geometry   |   No Pictures

Directions:

Choose the best answer for each question. Each question has five choices: A, B, C, D, and E.

This mock emphasizes application, algebraic reasoning, functions, coordinates, numerical reasoning, statistics, probability, sequences, and modeling.

Geometry, solid geometry, and laboratory questions are intentionally excluded.

A calculator may be used where appropriate.

NUMERATION AND OPERATIONS

1. If x is a positive real number and x + 1/x = 5, what is x² + 1/x²?

A) 21

B) 23

C) 25

D) 27

E) 29

Answer: B) 23

2. Which expression is equivalent to (x − 3)² − (x + 3)²?

A) −12x

B) −6x

C) 0

D) 6x

E) 12x

Answer: A) −12x

3. A number is increased by 20% and then decreased by 20%. Compared with the original number, the final number is:

A) 4% greater

B) 4% smaller

C) Equal

D) 20% smaller

E) 40% smaller

Answer: B) 4% smaller

4. If a/b = 3/5 and b/c = 10/7, what is a/c?

A) 3/7

B) 5/7

C) 6/7

D) 7/6

E) 15/7

Answer: C) 6/7

5. The greatest common factor of 84 and 126 is:

A) 14

B) 21

C) 28

D) 42

E) 63

Answer: D) 42

6. If n is an integer and n² is divisible by 12, which statement must be true?

A) n is divisible by 2 only.

B) n is divisible by 3 only.

C) n is divisible by 6.

D) n is divisible by 12.

E) n is prime.

Answer: C) n is divisible by 6.

7. What is the remainder when 7²⁰ is divided by 6?

A) 0

B) 1

C) 2

D) 4

E) 5

Answer: B) 1

8. If a = 2⁵ × 3² and b = 2³ × 3⁴, what is the greatest common factor of a and b?

A) 24

B) 36

C) 72

D) 108

E) 216

Answer: C) 72

ALGEBRAIC EXPRESSIONS, EQUATIONS AND INEQUALITIES

9. If 3x − 7 = 2x + 5, what is the value of x?

A) −12

B) −2

C) 2

D) 12

E) 14

Answer: D) 12

10. The equation x² − 8x + k = 0 has exactly one real solution. What is k?

A) 4

B) 8

C) 12

D) 16

E) 32

Answer: D) 16

11. If x + y = 9 and xy = 14, what is x² + y²?

A) 53

B) 55

C) 57

D) 67

E) 95

Answer: A) 53

12. Which value of x satisfies |2x − 5| = 9?

A) 2 only

B) −2 only

C) 7 only

D) −2 or 7

E) −7 or 2

Answer: D) −2 or 7

13. For which values of x is (x − 4)(x + 2) < 0?

A) x < −2

B) x > 4

C) −2 < x < 4

D) x ≤ −2 or x ≥ 4

E) −4 < x < 2

Answer: C) −2 < x < 4

14. If 1/x + 1/(x + 2) = 1, which value of x is valid?

A) −1

B) 0

C) 1

D) 2

E) 3

Answer: C) 1

15. If x² − 5x − 14 = 0, what is the sum of the reciprocals of the two roots?

A) −5/14

B) −1/2

C) 1/2

D) 5/14

E) 14/5

Answer: D) 5/14

16. If x − 1 is a factor of x³ + ax² − 5x + 3, what is a?

A) −3

B) −2

C) −1

D) 1

E) 3

Answer: C) −1

17. If x² + kx + 16 is a perfect square, which value could k have?

A) 2

B) 4

C) 6

D) 8

E) 10

Answer: D) 8

18. The equation 2x² − 3x − 5 = 0 has roots r and s. What is r² + s²?

A) 9/4

B) 13/4

C) 17/4

D) 29/4

E) 49/4

Answer: D) 29/4

FUNCTIONS AND MODELING

19. Let f(x) = 2x² − 3x + 1. What is f(−2)?

A) 3

B) 7

C) 11

D) 15

E) 19

Answer: C) 15

20. If f(x) = 3x − 2 and g(x) = x², what is f(g(2))?

A) 4

B) 8

C) 10

D) 12

E) 14

Answer: C) 10

21. A function is defined by f(x) = (x − 4)/(x + 2). Which value is excluded from its domain?

A) −4

B) −2

C) 0

D) 2

E) 4

Answer: B) −2

22. If f(x) = x² − 6x + 8, for which value of x is f(x) = 0?

A) 1

B) 2

C) 3

D) 4

E) 6

Answer: B) 2

23. Which function has an x-intercept at x = 3 and x = −5?

A) x² + 2x − 15

B) x² − 2x − 15

C) x² + 8x + 15

D) x² − 8x + 15

E) x² + 15

Answer: B) x² − 2x − 15

24. A linear function satisfies f(2) = 7 and f(5) = 16. What is f(10)?

A) 25

B) 28

C) 31

D) 34

E) 37

Answer: C) 31

25. If f(x) = x + 3 and f(g(x)) = 2x + 7, what is g(x)?

A) 2x + 3

B) 2x + 4

C) 2x + 7

D) 2x + 10

E) x + 4

Answer: B) 2x + 4

26. The function f(x) = −2(x − 3)² + 8 has its maximum value at:

A) −2

B) 0

C) 3

D) 6

E) 8

Answer: C) 3

27. If f(x) = 2ˣ and g(x) = x + 1, what is f(g(3))?

A) 8

B) 12

C) 16

D) 32

E) 64

Answer: C) 16

28. A population is modeled by P(t) = 500(1.04)ᵗ. What does 1.04 represent?

A) A decrease of 4% per period

B) An increase of 4% per period

C) An increase of 104% per period

D) The initial population

E) The number of periods

Answer: B) An increase of 4% per period

COORDINATE SYSTEM

29. A line passes through (2, 5) and (6, 13). What is its slope?

A) 1/2

B) 1

C) 2

D) 3

E) 4

Answer: C) 2

30. Which equation represents a line perpendicular to y = 3x − 7?

A) y = 3x + 4

B) y = −3x + 4

C) y = 1/3x + 4

D) y = −1/3x + 4

E) y = −4x + 3

Answer: D) y = −1/3x + 4

31. The line 2x + 3y = 12 has a y-intercept of:

A) 2

B) 3

C) 4

D) 6

E) 12

Answer: C) 4

32. If the point (a, 7) lies on the line y = 2x + 1, what is a?

A) 2

B) 3

C) 4

D) 5

E) 6

Answer: B) 3

33. The equation x² + y² − 6x + 4y − 12 = 0 represents a circle. What is its radius?

A) 3

B) 4

C) 5

D) 6

E) 7

Answer: C) 5

34. A parabola is given by y = (x − 2)² − 9. What is its minimum y-value?

A) −11

B) −9

C) −7

D) 2

E) 9

Answer: B) −9

35. The points (1, 4), (3, 10), and (5, 16) lie on the same line. Which equation represents the line?

A) y = 2x + 2

B) y = 3x + 1

C) y = 3x + 2

D) y = 4x

E) y = 4x + 1

Answer: C) y = 3x + 1

36. If a line has equation y = −4x + 9, which point lies on the line?

A) (1, 3)

B) (2, 1)

C) (3, 0)

D) (4, −6)

E) (5, −9)

Answer: B) (2, 1)

SEQUENCES, SERIES AND VECTORS

37. An arithmetic sequence begins 7, 12, 17, 22, ... What is the 20th term?

A) 92

B) 97

C) 102

D) 107

E) 112

Answer: C) 102

38. The first term of a geometric sequence is 3 and the common ratio is 2. What is the 8th term?

A) 192

B) 256

C) 384

D) 512

E) 768

Answer: C) 384

39. The sum of the first n positive integers is 210. What is n?

A) 15

B) 18

C) 20

D) 21

E) 24

Answer: D) 20

40. A geometric sequence has first term 81 and common ratio 1/3. What is the sum of its first four terms?

A) 108

B) 117

C) 120

D) 126

E) 135

Answer: D) 120

41. If the vectors u = (3, −2) and v = (−5, 4), what is 2u + v?

A) (1, 0)

B) (1, −8)

C) (6, 0)

D) (11, −8)

E) (11, 0)

Answer: A) (1, 0)

42. If the nth term of a sequence is aₙ = 3n − 2, what is a₁ + a₂ + a₃?

A) 9

B) 12

C) 15

D) 18

E) 21

Answer: C) 15

STATISTICS AND DATA ANALYSIS

43. The data set is 4, 7, 7, 9, 13. What is the median?

A) 4

B) 7

C) 8

D) 9

E) 13

Answer: B) 7

44. The mean of five numbers is 18. Four of the numbers are 12, 17, 21, and 24. What is the fifth number?

A) 14

B) 15

C) 16

D) 17

E) 18

Answer: C) 16

45. Which measure is least affected by an extreme outlier?

A) Mean

B) Median

C) Standard deviation

D) Variance

E) Range

Answer: B) Median

46. A data set has a minimum of 12 and a maximum of 47. What is its range?

A) 25

B) 30

C) 35

D) 45

E) 59

Answer: C) 35

47. A data set has Q1 = 14 and Q3 = 31. What is its interquartile range?

A) 15

B) 16

C) 17

D) 18

E) 45

Answer: C) 17

48. Two data sets have the same mean. Set A has standard deviation 2, while Set B has standard deviation 8. Which statement is correct?

A) Set A has greater spread.

B) Set B has greater spread.

C) Both have the same spread.

D) Standard deviation cannot measure spread.

E) Set B must have a greater mean.

Answer: B) Set B has greater spread.

49. A line of best fit is y = 2.5x + 4. What is the predicted value of y when x = 8?

A) 20

B) 22

C) 24

D) 26

E) 28

Answer: C) 24

50. A correlation coefficient of r = −0.92 indicates:

A) A strong positive linear relationship.

B) A weak positive linear relationship.

C) Almost no linear relationship.

D) A strong negative linear relationship.

E) A perfect nonlinear relationship.

Answer: D) A strong negative linear relationship.

PROBABILITY AND COUNTING

51. A fair die is rolled twice. What is the probability that both results are even?

A) 1/6

B) 1/4

C) 1/3

D) 1/2

E) 2/3

Answer: B) 1/4

52. A bag contains 5 red balls and 3 blue balls. Two balls are selected without replacement. What is the probability that both are red?

A) 5/14

B) 5/16

C) 10/28

D) 25/64

E) 1/2

Answer: A) 5/14

53. How many different arrangements can be made using the letters A, B, C, and D exactly once?

A) 4

B) 8

C) 12

D) 16

E) 24

Answer: E) 24

54. From 8 students, how many different groups of 3 students can be selected?

A) 24

B) 32

C) 56

D) 64

E) 336

Answer: C) 56

55. A probability experiment has P(A) = 0.4 and P(B) = 0.5. If A and B are mutually exclusive, what is P(A or B)?

A) 0.1

B) 0.2

C) 0.4

D) 0.5

E) 0.9

Answer: E) 0.9

56. A fair coin is tossed 4 times. What is the probability of getting exactly 3 heads?

A) 1/16

B) 1/8

C) 3/16

D) 1/4

E) 3/8

Answer: C) 1/4

HARD EST-STYLE MIXED REASONING

57. Let x and y be positive numbers such that x + y = 10. What is the greatest possible value of xy?

A) 20

B) 24

C) 25

D) 30

E) 50

Answer: C) 25

58. A quantity increases by 10% each year. If its initial value is 800, which expression represents its value after t years?

A) 800 + 10t

B) 800(0.10)ᵗ

C) 800(1.10)ᵗ

D) 810ᵗ

E) 800t¹·¹

Answer: C) 800(1.10)ᵗ

59. If x is a real number satisfying x + 1/x = 4, what is x³ + 1/x³?

A) 40

B) 48

C) 52

D) 56

E) 64

Answer: C) 52

60. The equation x² − px + 16 = 0 has two positive integer roots. Which value could p be?

A) 5

B) 7

C) 8

D) 9

E) 17

Answer: D) 9